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Integral of $$$\sin{\left(\nu \right)} \cos{\left(\mu \right)}$$$ with respect to $$$\mu$$$
Related calculator: Definite and Improper Integral Calculator
Your Input
Find $$$\int \sin{\left(\nu \right)} \cos{\left(\mu \right)}\, d\mu$$$.
Solution
Apply the constant multiple rule $$$\int c f{\left(\mu \right)}\, d\mu = c \int f{\left(\mu \right)}\, d\mu$$$ with $$$c=\sin{\left(\nu \right)}$$$ and $$$f{\left(\mu \right)} = \cos{\left(\mu \right)}$$$:
$${\color{red}{\int{\sin{\left(\nu \right)} \cos{\left(\mu \right)} d \mu}}} = {\color{red}{\sin{\left(\nu \right)} \int{\cos{\left(\mu \right)} d \mu}}}$$
The integral of the cosine is $$$\int{\cos{\left(\mu \right)} d \mu} = \sin{\left(\mu \right)}$$$:
$$\sin{\left(\nu \right)} {\color{red}{\int{\cos{\left(\mu \right)} d \mu}}} = \sin{\left(\nu \right)} {\color{red}{\sin{\left(\mu \right)}}}$$
Therefore,
$$\int{\sin{\left(\nu \right)} \cos{\left(\mu \right)} d \mu} = \sin{\left(\mu \right)} \sin{\left(\nu \right)}$$
Add the constant of integration:
$$\int{\sin{\left(\nu \right)} \cos{\left(\mu \right)} d \mu} = \sin{\left(\mu \right)} \sin{\left(\nu \right)}+C$$
Answer
$$$\int \sin{\left(\nu \right)} \cos{\left(\mu \right)}\, d\mu = \sin{\left(\mu \right)} \sin{\left(\nu \right)} + C$$$A